ABSTRACT
In this section, we look at the most successful lagrangian methodfor doing incompressible fluid simulation, known as Smoothed Particle Hydrodynamics, or SPH for short. As the name implies, the fluid is represented by a large
collection of individual fluid particles. These interact with each other following the
laws of hydrodynamics-as captured by the Navier-Stokes equations. Figure 14.1
shows what a typical SPH water simulation might look like if the water parti-
cles were rendered as colored spheres. Note that the simulation is especially good
at capturing detail in the splashing fluid motion. SPH found its original applica-
tions in the astrophysics community, with its underlying mathematical and physical
elaboration developed in a paper by Monaghan [1992]. The SPH method was first
introduced to the graphics community by Stam and Fiume [1995] for simulating
fire, and soon after was used by Desbrun and Gascuel [1996] for simulating highly
deformable objects. The paper that firmly established SPH as a method for fluid
simulation was by Mu¨ller et al. [2003], entitled Particle-Based Fluid Simulation for
Interactive Applications. As the title suggests, the original intent of SPH was to
provide interactive fluid-like animation for applications like VR and video games.
14.1 SPATIAL SAMPLING AND RECONSTRUCTION The basic idea of SPH is that the fluid mass is sampled onto a large system of
interacting particles, so that each particle can be thought of as representing a small
“chunk” or volume of the fluid. If the volume represented by particle i is Vi, centered
at position xi, its mass will be
